Study of the stress-strain and temperature fields in cutting tools using laser interferometry

OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 4 According to Hooke’s Law, ( ) , z x y E     = - + (4) where μ is the Poisson ratio, and E is the Young’s modulus, and σ x and σ y are the normal stresses along the X - and Y -axis directions, respectively. Using equation (1) for the case when there is only force action ( m = m p ) with equations (2) and (4), and considering ( σ x + σ y ) = Θ , we can obtain an equation for calculating the sum of the normal stresses Θ as: . p c c c m t E E t t      = - = - (5) In this way, the interference pattern analysis allows us to determine the fringe order differences at the point of interest in the cutting part of the tool, and using this information, determine the sums Θ of the stress components at this point. Before calculating the stress components, the fi eld of the sums Θ obtained from the experiment should be harmonized. The harmonization of the sums fi eld Θ is performed by solving the differential equation of equilibrium: 2 2 2 2 0. x y   ¶ ¶ + = ¶ ¶ (6) Equation (6), in fi nite difference form for a square grid (Fig. 3, a ), has the following form: 1, , 1 1, , 1 , 4 0, J N J N J N J N J N      + + - - + + + - ⋅ = (7) and for harmonization purposes, it can be transformed into: , 1. . 1 1, , 1 1 ( ). 4 J N J N J N J N J N      = + - - = + + + (8) After harmonization of the sums Θ fi eld by the iteration method to the required accuracy, the compo- nents of stresses  x ,  y , and  xy may be calculated. For the stress calculation, the location of the grid x -axis along the clearance face of the cutting tool is most convenient when it is formed as a wedge (Fig. 4). In orthogonal cutting conditions, the tool is in a plane stress state, which must satisfy the equilibrium equations: ô ó 0, xy x X x y ¶ ¶ + + = ¶ ¶ (9) а b Fig. 3. Scheme to number the nodes of the square grid ( a ) and scheme to end nodes of grid lines ( б )